Azula and Zuko adopted by Iroh
Basically Azulon order to ozai that he killed zuko was a test (in the fic he asked hin to kill azula) so azula come crying to zuko about it, and later they confronted ozai about it , him trying to kill his daughter/son caused azulon to banish him and putting him out of the line of sucession, the thing is that while this make thw fire family more stable caused this to make them anti-villains who sre full on board with conquering the world with azula being happy to be zuko second in command.
Azulon didn't just banish Ozai, he completely disowned him and struck him from any historical records about the fire nation family, changing history to say that Ursa married Iroh after he lost his first wife and that Zuko and Azula were their kids. Ozai, infuriated by this, slips into the background and begins plotting his revenge, planning to take over the fire nation and kill Azulon, Iroh, and Ursa. Fast forward, things are as you said in the fire nation family, with Azulon still on the thrown, until the events at the end of season one roll around. That's when Ozai makes his move. He blames the loss there on Iroh, using his opposition to Zhao, witnessed by several other members of the army, to make him out as a traitor who got Ozai cast out years earlier to prevent him from taking the thrown. He mounts a coup, kills Azulon and forces Iroh, Ursa, Azula, and Zuko into exile to escape him.